Noncommutative splitting fields

Abstract

For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits into linear factors; such an N is called a splitting field for p.(3) This extension N is unique.In the noncommutative case also a version of (1) can be proved; see forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin the noncommutative case no criterion on separability of p is needed; so,for instance we can construct noncommutative fields containing complete families of separate zeros for inseparable polynomials over commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.

abstract = "For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits into linear factors; such an N is called a splitting field for p.(3) This extension N is unique.In the noncommutative case also a version of (1) can be proved; see forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin the noncommutative case no criterion on separability of p is needed; so,for instance we can construct noncommutative fields containing complete families of separate zeros for inseparable polynomials over commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.",

N2 - For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits into linear factors; such an N is called a splitting field for p.(3) This extension N is unique.In the noncommutative case also a version of (1) can be proved; see forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin the noncommutative case no criterion on separability of p is needed; so,for instance we can construct noncommutative fields containing complete families of separate zeros for inseparable polynomials over commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.

AB - For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits into linear factors; such an N is called a splitting field for p.(3) This extension N is unique.In the noncommutative case also a version of (1) can be proved; see forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin the noncommutative case no criterion on separability of p is needed; so,for instance we can construct noncommutative fields containing complete families of separate zeros for inseparable polynomials over commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.